Read noise increases with ISO, because any noise added from the sensor readout before ISO amplification is then amplified by the ISO gain amplifier. A crude model of the variation of read noise with ISO is thus to assume independent pre- and post-amplification noise sources, that are combined as usual in quadrature:

R= (G R)+ R

In terms of this discussion, we can divide sensors into two types: ISOless and non-ISOless. ISOless sensors are sensors where the read noise remains the same regardless of the amplification (e.g. the sensor in the D800). In contrast, non-ISOless sensors have less read noise as the amplification increases compared to using base ISO and pushing.

For example, the sensor in my Canon 6D is not ISOless. That means a photo at f/2.8 1/100 ISO 1600 will be less noisy than a photo of the same scene at f/2.8 1/100 ISO 100 pushed four stops in the RAW conversion, despite having the same exposure (recall that the ISO setting has nothing to do with exposure), whereas it would be the same noise for a camera with an ISOless sensor.

So, for sure f/2.8 1/1600 ISO 1600 will be more noisy than f/2.8 1/100 ISO 100 -- not because of the higher ISO setting, but because the shutter speed is 16x faster, resulting in 1/16 as much light falling on the sensor, and thus 4x the photon noise.

But, f/2.8 1/100 ISO 100 pushed four stops in the RAW conversion will have the same noise as f/2.8 1/100 ISO 1600 using an ISOless sensor, and more noisy if using a non-ISOless sensor.

In other words, it is in that context that I say higher ISOs are less noisy than lower ISOs for a given exposure if using a camera with a non-ISOless sensor.

Regardless, this is neither here nor there with regards to Equivalence. If we shoot a scene at f/2.8 1/100 on 1.6x and f/4.5 1/100 on FF the same total amount of light will fall on the sensor, so the photon noise will be the same if the sensors have the same QE (Quantum Efficiency -- the proportion of photons falling on the sensor that are converted to electrons). This is true regardless of the ISO setting or pixel count. It is also worthwhile to note that sensors of the same generation have remarkably consistent QEs, even across brands.

However, while the photon noise is dominant for the majority of most photos, the less light that makes up a portion of a photo, the more the read noise (the additional noise added by the sensor and supporting hardware) becomes a factor.

It is here that the pixel count and ISO setting have a secondary effect on noise. For an ISOless sensor where the read noise were proportional to the pixel pitch, then neither the pixel size nor the ISO setting would have any effect. As it turns out, the reality is closer to sensors of the same generation having about the same read noise per pixel, so more smaller pixels will be more noisy than fewer larger pixels. But this will only become apparent in portions of the photo receiving very little light.

For example, we can see that the Canon 6D and the Canon 70D have almost the same QE and, at higher ISO settings, the same read noise per pixel (at lower ISOs, the read noise is insignificant compared to the photon noise). Thus, for example, f/2.8 1/100 ISO 800 on the 70D will have pretty much the exact same noise as f/4.5 1/100 ISO 2000 on the 6D.

So, let's recap: the primary sources of noise in a photo are the photon noise (noise from the light itself) followed by the read noise (noise from the sensor and supporting hardware).

The photon noise is determined by the total amount of light falling on the sensor and the QE (Quantum Efficiency -- the proportion of photons that are converted into electrons -- which is remarkably consistent, even across brands, for sensors of the same generation).

The read noise, however, varies considerably, even for sensors of the same generation, at lower amplifications, but is more consistent at higher amplifications. Thus, sensors with more pixels tend to have more noise for low light than sensors with fewer pixels, although typically not an issue until you are shooting at least around ISO 3200.